arXiv:1807.02411v1 [math.CO] 5 Jul 2018 ojcue n hsteSalyWl ojcue acsa Marcus conjecture. Stanley-Wilf Ta the and thus Marcus and permutation. conjecture, a of relati limit F¨uredi-Hajnal polynomial the a proved and later [8] Cibulka respectively. ifadFueiHja ojcue r aldthe called are F¨uredi-Hajnal conjectures and Wilf fpruainmtie slna.Kaa 7 hwda equi an constan The showed [7] F¨uredi-Hajnal conjecture. the Klazar and conjecture linear. is matrices t a that permutation avoiding states of permutations [6] of F¨uredi-Hajnal conjecture number The the ponential. of rate growth the independen Wilf that and Stanley st by the formulated in of conjecture, problems Wilf largest area the the of One to [4]. introduction Knuth the to attributed [3], Kitaev extr to the According of behavior [2]. asymptotic [1] the substructure describing specific is a avoidance containing without have a can often avoidance ture pattern of theory extremal the in Problems Background 1.1 Introduction 1 2 on fteetea ucinof function extremal the multid associated of their bound of function extr extremal the the bound of to terms method in new seve a develop prove We to graphs. equivalence of the functions graph use bipartite We of class function. extremal a trix for function extremal equi graph an prove the We of generalizations. its and function extremal O ( k udmna rbe nptenaodnei eciigth describing is avoidance pattern in problem fundamental A mrvdbud nteetea ucinof function extremal the on bounds Improved ) n d − 1 . hypergraphs d ila Zhang William pruainhprrpso length of hypergraphs -permutation Abstract 1 tne-iflimit Stanley-Wilf egot aeo h xrmlfunction extremal the of rate growth he sin ts nhpbtenteSalyWl limit Stanley-Wilf the between onship d fptenaodnei h Stanley- the is avoidance pattern of udy mninlmtie,ipoigthe improving matrices, imensional kfrtenme feeet struc- a elements of number the for sk mlfnto n t generalizations its and function emal l ntelt 90 5,wihstates which [5], 1980s late the in tly udmna rbe npattern in problem fundamental A . n h smttc ftema- the of asymptotics the and s aec ewe h asymptotics the between valence emtto atrsi traditionally is patterns permutation a e onso h extremal the on bounds new ral dTro’ okhv ic then since have work Tardos’s nd ds[]poe h F¨uredi-Hajnal the proved [9] rdos aec ewe h Stanley-Wilf the between valence 2 mlfnto fhypergraphs of function emal O ie emtto atr sex- is pattern permutation given smttcbhvo fthe of behavior asymptotic e ( n ) n the and and k O from ( n F ) rd-anllimit uredi-Hajnal ¨ rmteStanley- the from O ( n d − 1 ) to , been generalized in various directions, and many others have significantly sharpened their bounds [10, 11, 12, 13, 14, 15, 2]. Following the Marcus-Tardos Theorem, Fox [10] proved exponential upper and lower bounds on the Stanley-Wilf limit, disproving a widely believed conjecture that the Stanley-Wild limit was quadratic in the length of its permutation [16]. This problem has been extensively studied both out of mathematical interest and due to its applications in computational geometry and engineering. Mitchell’s algorithm [17] computes a shortest rectilinear path avoiding rectilinear obstacles in the plane. Mitchell showed that the com- plexity of the algorithm can be bounded by the extremal function of a specific . Bienstock and Gy¨ori [18] bounded the complexity of the algorithm by finding sharp upper bounds on the extremal function of that matrix. Mitchell’s algorithm has direct applications in both motion plan- ning in robotics and wire routing in VLSI circuit design [19]. Furthermore, F¨uredi [20] used the extremal function to find an upper bound on the Erd˝os-Moser problem [21] of determining the maximum number of unit distances in a convex polygon. Aggarwal [22] sharpened F¨uredi’s re- sult on the upper bound on the maximum number of unit distances in a convex polygon. Some problems from pattern avoidance also emerged in bounding the number of possible lower envelope sequences formed by continuous functions [23].

1.2 Definitions

We denote the list {1,...,n} = [n].

For some integer d ≥ 2, a d-dimensional matrix is a block of numbers on [n1] ×···× [nd]. We denote such a matrix A = (A(i1,...,id)) where 1 ≤ il ≤ nl for each l ∈ [d]. In this paper we only consider binary matrices, so every entry is a 0-entry or a 1-entry. When we refer to a d-dimensional matrix A having a side length k, we mean that A is a block of numbers on [k]d. We also refer to a d-dimensional matrix as a d-matrix. We denote the number of 1-entries in a d-matrix A by w(A). th An l-cross section of matrix A is the set of all the entries A(i1,...,id) whose l coordinates have

the same value. An l-row of matrix A is the collection of all the entries A(i1,...,id) whose coordinates other than the lth coordinate have fixed values.

Definition. Let π1,...,πd−1 be permutations on [k]. Then the matrix A that is defined by

A(i,π1(i),...,πd−1(i)) = 1 for each i ∈ [d − 1] and 0-entries everywhere else is the d-permutation

matrix of length k constructed from π1,...,πd−1. A refers to a 2-permutation matrix. Definition. If A and B have the same dimensions and B can be obtained from A by changing 1-entries in A into 0’s, then A represents B. If some submatrix of A represents B, then A contains B. Otherwise, A avoids B. The extremal function ex(B, n) denotes the maximum possible number of 1-entries in an n×n binary matrix that

2 avoids B. We call B linear if ex(B, n) = Θ(n). For d-dimensional matrices, f(B,d,n) denotes the maximum possible number of 1-entries in a d-matrix of side length n that avoids B. An ordered hypergraph is an ordered pair H =(V, E) where V is a linearly ordered set and E is a set of subsets of V . Each v ∈ V is a vertex of H, and each e ∈ E is an edge of H. The weight N of a hypergraph H =(V, E) is e∈E |e|. For d ∈ , a hypergraph H =(V, E) is d-uniform if for each e ∈ E,we have |e| = d. AnPordered graph is a 2-uniform ordered hypergraph. Because this paper does not deal with unordered graphs and unordered hypergraphs,we refer to ordered graphs and ordered hypergraphs as just graphs and hypergraphs.

Definition. A hypergraph A =(V1, E1) contains another hypergraph B =(V2, E2) if there exists an increasing injection f : V2 7→ V1 and an injection g : E2 7→ E1 such that for each e ∈ E2,we have that f(e) ⊂ g(e). Otherwise, A avoids B. If f and g are bijections such that f(e)= g(e) for each e ∈ E2, then A and B are order-isomorphic. If G is a graph, the extremal function for graphs gex(G, n) denotes the maximum possible number of edges in a graph with n vertices such that A avoids G. Analogously,we associate two

extremal functions for hypergraphs. If H is a hypergraph, then exe(H, n) denotes the maximum

possible number of edges of a hypergraph on [n] that avoids H, and exi(H, n) denotes the maxi- mum possible weight of a hypergraph on [n] that avoids H.

Definition. Let H =(V, E) be a hypergraph with V = {v1,...,vn} such that v1 < ··· < vk. Ifwe

can partition V into d sets Ii = {vki−1+1,...,vki } for i ∈ [d] with k0 = 0 and kd = n such that

each e ∈ E contains at most one vertex from each Ii, then H is d-partite. Each Ii is a part of H.

Definition. Given a d-dimensional matrix M with dimensions [n1] ×···× [nd], the hypergraph d associated with M is H = ([ i=1 ni], E) where for each 1-entry mk1,...,kd , E contains the edges − j 1 m k d . Conversely, for a d-partite, d-uniform hypergraph H′, the d-matrix associ- {( i=1 i)+ j}j=1 P H′ d M ′ H′ M ′ atedP with is the -matrix such that is associated with . We see that if M has side length n, then H is a d-partite graph on nd vertices with each part of size n. A d-permutation hypergraph of length k ∈ Z+ is a d-uniform, d-partite hypergraph H = ([kd], E) with parts of size k such that each vertex v ∈ [kd] is in exactly one edge. Similarly, a permutation graph of length k is the graph associated with a 2-permutation matrix. We see that every d-permutation hypergraph is the hypergraph associated with a d-permutation matrix, and vice versa. Klazar and Marcus [11] observed that if G and G are d-partite, d-uniform hypergraphs with nd vertices and parts of size n, then G contains H if and only if the the matrix associated with G contains the matrix associated H.

Definition. In a d-matrix P , the distance vector between entries P(a1,...,ad) and P(b1,...,bd) is (b1 − d d a1,...,bd − ad) ∈ Z . A vector x ∈ Z is r-repeated in a permutation matrix P if x occurs as the distance vector of at least r pairs of 1-entries.

3 1.3 New Results

We prove several new bounds on the extremal functions of graphs and multidimensional matrices using techniques from the extremal theory of matrices, probability, and analysis. We also develop new methods for bounding the extremal function of hypergraphs in terms of the extremal function of multidimensional matrices. In section 2, we prove an equivalence between the asymptotics of the graph extremal func- tion for a class of bipartite graphs and the asymptotics of the matrix extremal function. We use the equivalence as well as upper bounds obtained from Cibulka and Kyn˘cl [12] to prove that 8 2 4(k+1) gex(P, n) ≤ 3 (k + 2) 2 n for all permutation graphs P of length k. We use the equiva- lence to improve the known upper bound for j-tuple permutation graphs to gex(P, n)=2O(k)n. The previous bound proven by Weidert [15] was gex(P, n)=2O(k log k)n. We also generalize the upper bound 2O(k2/3(log k)7/3)/(log log k)1/3 n for the extremal function of almost all permutations matrices [12] to the extremal function of almost all permutation graphs. In section 3, we generalize the upper bound on graphs in Lemma 2.2 to hypergraphs. For a d−1 d-permutation hypergraph P of length k, we improve the bound exi(P, n) = O(n ) obtained O(k) d−1 by Gunby and P´alv¨olgyi [2] to exi(P, n)=2 n . This also generalizes Geneson and Tian’s result [13] that f(Q, d, n)=2O(k)nd−1, where Q is a d-permutation matrix of length k. We also sharpen Lemma 7.1 of [2] by bounding the number of hypergraphs avoiding a given d-permutation hypergraph to 22O(k)n. Furthermore, our proof extends to when P is the hypergraph associated with a j-tuple d-permutation matrix of length k. In section 4, we use the probabilistic method to derive lower bounds for the extremal functions mentioned in this paper. We generalize a lower bound of a completely filled matrix [13] to a lower bound on arbitrary matrices and graphs. Crowdmath [24] proved that for an r × c binary matrix B, if it has more than r + c − 1 one entries, then ex(B, n) = Ω(n log n). We use the new lower bound to show that if B has more than r + c − 1 one entries, then ex(B, n) = Ω(n1+ǫ) for some ǫ > 1. We also generalize this lower bound to arbitrary hypergraphs. Furthermore, we use the

lower bounds for f(P,d,n) for d-permutation matrices [13] to find lower bounds on exi(Q, n) where Q is a d-permutation hypergraph. This lower bound shows that our upper bound for the hypergraph extremal function of d-permutations is tight up to a constant dependent on d.

2 Equivalence of graph and matrix extremal functions

In this entire section, unless otherwise stated, let P be a matrix on [k1] × [k2] with a 1-entry at

P(k1,1). Let Q be the graph associated with P .

Theorem 2.1. gex(Q, n) ∼ ex(P, n).

4 We generalize Corollary 2.2.9 from [14].

Lemma 2.2. For all n ∈ Z+ gex(Q, n) ≤ ex(P, n).

Proof. Let A = ([n], E) be a graph avoiding Q. Let B be the n × n matrix defined by Bij = 1 if {i, j} ∈ E and i < j, and let C = ([2n], E′) be the graph associated with B. The number of 1-entries in B is |E|. We also have {i, j} ∈ E if and only if {i, j + n} ∈ E′. Suppose for contradiction that B has more than ex(P, n) 1-entries. Then B contains P . Let P ′ be the ′ submatrix of B that represents P , where the rows of P are {r1,...,rk1 } ⊂ [n] and the columns ′ of P are {c1,...,ck2 } ⊂ [n] with r1 < ··· < rk1 and c1 < ··· < ck2 . Since the bottom-left ′ ′ 1-entry of P is a 1-entry in B, by construction of B, we have rk1 < c1. Let B be the graph associated with P ′, so Q is contained in B′, which is contained in C. Let G = (V, F ) be the ′ copy of Q in B . Then V = {r1,...,rk1 , n + c1,...,n + ck2 }. If f is the increasing injection

from {r1,...,rk1 , n + c1,...,n + ck2 } to {r1,...,rk1 ,c1,...,ck2 }, then f(G)= Q, so G is order- isomorphic to Q. Then since A contains G, we have that a contains Q, contradiction.

Also note that we can use a symmetrical argument if P has a 1-entry in the top-right corner.

Lemma 2.3. For all n, t ∈ Z+, we have gex(Q, nt) ≥ (t − 1)ex(P, n).

Proof. Let A be a bipartite graph on [2n] with parts {1,...,n} and {n +1,..., 2n} that avoids Q with ex(P, n) edges. Let I,J ⊂ [n] such that the edges of A are {i, n + j} for i ∈ I and j ∈ J. Let G be a graph with vertex set [nt] and edges {(k − 1)n + i, kn + j} for each i, ∈ I, j ∈ J, and

k ∈ [t − 1]. We show that G avoids Q. Define intervals Ik = [(k − 1)n +1, kn] for each k ∈ [t]. We see that every edge in G connects vertices in consecutive intervals. For contradiction, suppose G′ = (V ′, E′) is subgraph of G isomorphic to Q, so G′ must ′ also be bipartite. Let the parts be V1 and V2. Suppose G contains vertices from three intervals ′ ′ Ix−1,Ix,Ix+1. Let f :[k1 + k2] 7→ V be the isomorphism from Q to G . Without loss of general-

ity, suppose f({k1,k1 +1}) = {v1, v2} such that v1 ∈ Ix−1 and v2 ∈ Ix. Then V1 ⊂ Ix−1. Since ′ there are no vertices in Ix+1 adjacent to any vertices in Ix−1, it follows that V2 ⊂ Ix. If G contains ′ vertices from only two different intervals Ix and Ix+1, then G is order-isomorphic to a subgraph of A, so then G′ avoids Q.

Now we prove the main theorem of this section.

ex(P,n) R Proof. From [10], we have limn→∞ n = cP for some cP ∈ . Then ex(P, n) = cP n + o(n). 3 3 2 Then from Lemma 2.2 and Lemma 2.3, ex(P, n ) ≥ gex(Q, n ) ≥ (n − 1)(cP n + o(n)), so 3 3 3 gex(Q,n) gex(Q, n )= n cP + o(n ), which implies limn→∞ n = cP .

5 Lemma 2.2 also has some corollaries that improve known bounds in other problems, specifi- cally, when P is a permutation matrix or a j-tuple permutation matrix. 8 2 4k Cibulka and Kyn˘cl [12] proved that ex(P, n) ≤ 3 (k +1) 2 n for all 2-permutation matrices P of length k. Appending a new row and a new column of P to obtain a (k +1)×(k +1) permutation matrix P ′ with a 1 in the bottom-left corner results in the following corollary:

8 2 4(k+1) Corollary 2.4. For all permutation graphs Q of length k, we have gex(Q, n) ≤ 3 (k +2) 2 n. Furthermore, this argument also extends the known bound of the graph extremal function of almost all permutation. Cibulka and Kyn˘cl [12] also proved that for almost all k × k permutation matrices that are r-repetition free, we have ex(P, n)=2O(r1/3k2/3(log k)2)n. If P is r-repetition free, then P ′ is (r + 1)-repetition free.

Corollary 2.5. For almost all permutation graphs Q with length k, we have gex(Q, n) = 2/3 7/3 1/3 2O((k (log k) )/(log log k) )n.

A j-tuple permutation matrix of length k is a k ×kj matrix that results from replacing each one entry in a permutation matrix with a 1×j matrix of ones and each zero entry with a 1×j matrix of zeros. Then the j-tuple permutation graph is a graph associated with a j-tuple permutation matrix. Geneson and Tian [13] proved that ex(P, n)=2O(k)n for all j-tuple permutation matrices P of 4 2k2 permutations of length k. We improve the bound gex(P, n) ≤ 11k 2k n from Corollary 3.0.6 of [15] with the same method.  Corollary 2.6. For all j-tuple permutation graphs Q of length k, we have gex(Q, n)=2O(k)n.

3 Improved upper bound on hypergraph extremal function

We improve the bound found by [2] and provide a more elegant argument by building off of the results of [13] and generalizing our Lemma 2.2.

Theorem 3.1. Let H be a fixed d-permutation hypergraph with k edges. Then exi(H, n) = 2O(k)nd−1.

We find a class of d-partite hypergraphs whose extremal functions can be bounded by the extremal functions for their associated d-matrices.

Lemma 3.2. For some t, d ∈ Z+, let H = ([dt],D) be a d-uniform d-partite hypergraph with parts of size t such that for each i ∈ [d − 1], H has an edge that is a superset of {it, it +1}. Let G = ([n], E) be a d-uniform hypergraph that avoids H. If P is the d-dimensional matrix associated with H, then the number of edges in G is at most f(P,d,n).

6 Proof. Let A be the d-dimensional matrix with side length n such that for each e = {k1,...,kd} ∈

E with k1 < ···

By the construction of A it follows that rit < rit+1. Then 1 ≤ r1 < ··· < rdt ≤ n. Then let ′′ ′ ′ ′ ′ H =(V , F ) be the graph defined by V = {r1,...,rdt} that contains edges {ri1 ,...,rid } ∈ F if A = 1. Clearly H′′ is contained in G, and H′′ is order-isomorphic to H. Then G (ri1 ,...,rid ) contains H, contradiction.

Lemma 3.3. Let H be a fixed d-permutation hypergraph of length k. Then there exists a d- permutation hypergraph H′ of length k + d − 1 that contains H and satisfies the conditions of Lemma 3.2.

Proof. Let P be the d-matrix of H, so P has side length k. Let Q be the d-permutation of length d that has one entries at each of the cyclic variants of (1,...,d); i.e. (1, 2,...,d), (d, 1,...,d − 1), ..., (2, . . . , d, 1). Construct P ′ by replacing the one entry at (1,...,d) with P , so P ′ has side length k + d − 1. Let H′ be the hypergraph associated with P ′. Clearly H′ is a d-permutation of length (k + d − 1) that contains H. For i = 2,...,d, the (d − i + 1)th coordinate of the entry (i, . . . , d, 1,...,i − 1) of Q is d. This entry corresponds to an entry in P ′ whose (d − i +1)th coordinate is k+d−1 and whose (d−i+2)th coordinate is 1. Then the edge in H′ that corresponds to that entry contains {(d − i + 1)(k + d − 1)+(k + d − 1), (d − i + 2)(k + d − 1)+1}.

Now we prove the main theorem of this section.

Proof. If H is a d-permutation hypergraph of length k, then Lemma 3.2, there exists a d- permutation hypergraph H′ = ([(k + d − 1)d],D) of length k + d − 1 that contains H and satisfies the conditions of Lemma 3.2. Let P ′ be the d-matrix associated with H′. Let G = ([n], E) be a hypergraph avoiding H, so G also avoids H′. Create G′ = ([n], E′) from G by removing every edge from E with size less than d. Create G′′ = ([n], E′′) from G′ by ′ replacing every edge having more than kd vertices {v1,...,vl} ∈ E with {v1,...,v(k+d)d}. For each edge in e ∈ E′′, there are at most d edges in E′ that map to e, otherwise G′ would contain H’.

For each i = d,d +1,...,kd, let Gi = ([n], Ei) be the i-uniform hypergraph that consists of every edge of G′′ of size i.

7 ′ ′ Let Pd−1 = P , and let Pi be a d-permutation matrix of length k + i that contains P and Pi−1

such that associated hypergraph of Pi satisfies the conditions of Lemma 3.2. We construct each Pi

by inserting a 1-entry somewhere in Pi−1 between any consecutive cross sections. ′ Let Hi be the d-permutation hypergraph associated with Pi. Since Hi contains H , Gi avoids

Hi. From Lemma 3.2, we see that |Ei| ≤ f(Pi,d,n). Since each Pi is contained in Pkd, it follows that f(Pi,d,n) ≤ f(Pkd,d,n).

n n |E| ≤ + ··· + + |E′| 0 d − 1     ≤ dnd−1 + d|E′′| kd d−1 ≤ dn + d |Ei| ! Xi=d kd d−1 ≤ dn + d f(Pkd,d,n) ! Xi=d ≤ dnd−1 + kd2 2O(kd)(n)d−1 − =2O(k)nd 1,  where the constant hidden in O(k) depends on d. We used the Theorem 4.1 from [13], which states that f(P,d,n)=2O(k)nd−1 for any d-permutation matrix P of length k. Then from [11], I have

exi(H, n) ≤ (2kd − 1)(k − 1)exe(H, n), so the result follows.

Geneson and Tian [13] showed that for any j-tuple d-permutation matrix of length k, I have f(P,d,n)=2O(k)nd−1. ItiseasytomodifytheproofofTheorem 3.1togetthefollowing corollary.

Corollary 3.4. If P is a hypergraph associated with a j-tuple d-permutation matrix of length k, O(k) d−1 then exi(P, n)=2 n .

Theorem 3.5. Let H be a d-permutation hypergraph of length k. The number of hypergraphs with vertex set [n] that avoid H is at most 22O(k)nd−1 .

Proof. Let M(H, n) be the set of hypergraphs on [n] that avoid H.

Let G ∈ M(H, tn). For each i ∈ [n], let the intervals Ii = {(i − 1)t +1,...,it}. Create a ′ ′ new graph G on [n] such that for each edge e of G, if e has vertices in Ik1 ,...,Ikl , then G has ′ ′ the edge {k1,...,kl}. Since G contains G , it follows that G also avoids H. Let f : M(H, tn) 7→ M(H, n) be the map from each G to G′. For each G′ ∈ M(H, n), at most (2t − 1)exe(H,n) graphs G ∈ M(tn, H) map to G′ because each incidence in G′ has 2t − 1 possible sets of incidences in ′ G that map to it. Then |M(tn, H)| ≤ (2t − 1)exi(H,n)|M(n, H)|. Let f(n) = log |M(n, H)|. We

8 get f(tn)= f(n)+2O(k) log (2t − 1)nd−1, so iterating this inequality gives l f(tl)= 2O(k) log (2t − 1)(ti−1)d−1. i=1 − X O(k) d−1 Then f(tl)=2O(k)(tl)d 1, so |M(n, H)| =22 n

4 Lower bounds on extremal functions

The following lemma is a generalization of Theorem 2.1 from [13].

Lemma 4.1. Let B be a d-matrix on [k1] ×···× [kd] . Then k +···+k −d d− 1 d f(B,d,n) = Ω n w(B)−1 .

Proof. Let A be a of side lengthn such that each entry is a 1 with probability p, and each entry is chosen independently of others. The expected value of w(A) is ndp. We form A′ as follows: from every k1 ×···× kd submatrix of A, if that submatrix represents B, then replace a one entry with a 0 so that the submatrix then avoids B. We see that A′ avoids B. Each submatrix represent B with probability pw(B). The expected number of ones in A′ is d ··· n (en)k1+ +kd E[w(A′)] = ndp − pw(B) ≥ ndp − pw(B), k1 k ki k ··· k d i=1  ! 1 d Y − k1+···+kd−d 1 w(B)−1 where we use the Stirling approximation for the inequality. Choose p = 2 n . There exists an event such that w(A′) ≥ E[w(A′)], so the result follows.

This proves a stronger condition of nonlinearity than the bounds shown by Crowdmath [24].

Corollary 4.2. If B is an r × c matrix with w(B) >r + c − 1, then ex(B, n)=Ω(n1+ǫ) for some ǫ> 0.

We can use the same method to bound the extremal function of all graphs.

2− k−2 Theorem 4.3. Let G = ([k], E) be a graph. Then gex(G, n)=Ω(n |E|−1 ).

Fox improved the probabilistic lower bound from [13], showing that for almost all d- permutation matrices P of length k, we have f(P,d,n)=2Ω(n1/2)nd−1. Klazar and Marcus [11] observed that if any d-matrix A avoids P , then the hypergraph associated with A avoids the hyper- graph associated with P . We use their observation to obtain the following statement.

Corollary 4.4. For almost all d-permutation hypergraphs P of length k, we have Ω(k1/2) d−1 exi(P, n)=2 n . Combining our upper and lower bounds shows that for almost all d-permutation hypergraphs kΘ(1) d−1 P of length k, we have exi(P, n)=2 n , indicating that our bounds are tight.

9 5 Conclusion

We reduced the calculation of the extremal function a class of bipartite graphs to the calculation of their associated matrices by showing an equivalence between the two problems. We bounded extremal function of d-permutation d-partite hypergraphs in terms of the extremal function of their associated d-matrices. We also obtained improved lower bounds for the extremal function of all d-matrices and graphs with the probabilistic method. One possible future direction for this research would be to show that f(P,d,n) = 2O(k2/3+o(1))nd−1 for almost all d-permutation matrices of length k. Using a similar method as the one used by [12], it seems likely that their argument can generalize to d> 2. This would also O(k2/3+o(1)) d−1 imply that exi(Q, n)=2 n for almost all d-permutation hypergraphs Q of length k. Another possible direction would be to apply our results to the extremal function of parti- tions, studied in [2]. Gunby and P´alv¨olgyi used the hypergraph extremal function to find doubly exponential upper bounds on the number of partitions that avoids a given pattern. We can use our improved result on the hypergraph extremal function to sharpen the bounds on the partition extremal function.

6 Acknowledgements

The author would like to thank Dr. Jesse Geneson for his valuable advice and guidance through the duration of this project. The research of the author was also supported by the Department of , MIT through PRIMES-USA 2017.

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